Time decay for nonlinear wave equations in two space dimensions

The Cauchy Problem for the equation u_tt–u+|u|^p–1u=0 (x², t>0, >1) is studied. Smooth Cauchy data is prescribed, and no smallness condition is imposed. For >5, it is shown that the maximum amplitude of such a wave decays at the expected rate t^–1/2 as t. For 1+8<5, the maximum amplitude still decays, but at a slower rate. These results are then used to demonstrate the existence of the scattering operator when >_o, where _o is the root of the cubic equation ³-2²-7-8=0; thus _o4.15.Alfred P. Sloan Research Fellow     

Boundary values of analytic operator functions with a positive imaginary part

Let be the Schatten-von Neumann class, 0<< of operators in a Hilbert space. It is shown that for-valued R-functions the nontangential boundary values exist almost everywhere on and belong to the class if 0<<1. For >1 the boundary values may be unbounded operators everywhere on . Finally, for =1 the nontangential limits exist in the norm of the space, for any q>1. However, they belong to a class which, in general, cannot be smaller than the class, conjugate with the Matsaev ideal.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 157, pp. 55–69, 1987.In conclusion, the author expresses his gratitude to M. Sh. Birman for his interest in the paper and for useful remarks.     

Some inequalities about the covering radius of Reed-Muller codes

Let R(r, m) be the rth order Reed-Muller code of length 2 ^m , and let (r, m) be its covering radius. We prove that if 2 k m - r - 1, then (r + k, m + k) (r, m + 2(k - 1). We also prove that if m - r 4, 2 k m - r - 1, and R(r, m) has a coset with minimal weight (r, m) which does not contain any vector of weight (r, m) + 2, then (r + k, m + k) (r, m) + 2k(. These inequalities improve repeated use of the known result (r + 1, m + 1) (r, m).This work was supported by a grant from the Research Council of Wright State University.     

Generalization of the theorem of Marcinkiewicz on entire characteristic functions of probability distributions

The relation is studied between the distribution of the zeros and the order of growth of entire analytic functions for which ¦p(z)¦ (i Imz) for Imz 0, in particular, of entire characteristic functions of probability distributions. The main result is the following: if ₁ is the exponent of convergence of the sequence of zeros of such a function of order which lie in a half plane Imz d > 0, then the inequality ₁ < implies the inequality p 3. This estimate is precise.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 85, pp. 94–103, 1979.The author is grateful to I. V. Ostrovskii for posing the problem and for his constant assistance with the work.     

Properties of solutions of linear functional-differential equations depending on a parameter

On the segment 0 t1 we study the equation A(d/dt, )x(t) + [F()x](t)=f(t), whereA (d/dt, ) x=x( ⁿ )+A ₁ x(^n–1 +...+ ⁿ A _n x, the matrices A₁,...,A_n are of size m × m, x is an unknown and f a given function with values in the m-dimensional space ^m , F() is a linear operator acting from a Hölder space to a Lebesgue space of vectorfunctions with values in ^m and depending on a complex parameter . We find the set of those at which a one-to-one correspondence is established between the solutions of the given equation and the solutions of the equation A(d/dt, )x(t)=0.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 9, pp. 1213–1231, September, 1991.     

Holomorphe Funktionen auf Gebieten über Banach-Räumen zu vorgegebenen Konvergenzradien

Given a function: ₊ on a domain spread over an infinite dimensional complex Banach space E with a Schauder basis such that -log is plurisubharmonic and d (d denotes the boundary distance on ) one can find a holomorphic function f: with _f, where _f is the radius of convergence of f. If, in addition, is locally Lipschitz continuous with constant 1, f can be chosen so that (3M)^–1 _f, where M is the basis constant of E. In the particular case of E= ₁ there are holomorphic functions f on with= _f.     

A Stationary Rho-Mixing Markov Chain Which Is Not “Interlaced” Rho-Mixing

A strictly stationary, countable-state Markov chain is constructed which is -mixing (with arbitrarily fast mixing rate) but fails to be *-mixing (interlaced-mixing).     

Best approximation and best unilateral approximation of the kernel of a biharmonic equation and optimal renewal of the values of operators

For the classB _p , 0 < 1, 1p , of 2-periodic functions of the form f(t)=u(,t), whereu (,t) is a biharmonic function in the unit disk, we obtain the exact values of the best approximation and best unilateral approximation of the kernel K(t) of the convolution f= K *g, gl, with respect to the metric of L₁. We also consider the problem of renewal of the values of the convolution operator by using the information about the values of the boundary functions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol.47, No. 11, pp. 1549–1557, November, 1995.     

Solvability of partial differential equations of infinite order in certain classes of entire functions

In this paper it is shown that under conditions of applicability of the operator to the class [,] =(I,s), ₂ 1, ₂), ₁, ₂< the equation y=f has a particular solution of this class vf[, ]. The general form of a solution of the homogeneous equation y=0 is established. The growth of a solution is investigated by means of a system of conjugate orders and a system of conjugate types. A solvability result is also obtained in the class , where T is a certain set in R ₊ ² depending on the operator .Translated from Matematicheskie Zametki, Vol. 19, No. 2, pp. 225–236, February, 1976.In conclusion, the author would like to express his thanks to his adviser, Yu. F. Korobeinik.     

Some properties of the delay probability inM/M/s/s +c systems

We look at an extension of the steady state delay probability inM/M/s/s + c systems to nonintegral number of serverss and queue capacityc, which we call GED function. We show that this function is increasing and concave in the queue capacity. We find that if c 1, the reciprocal of the GED function is convex in the traffic intensity and the GED function is increasing in the traffic intensity if is below some ^* _s,c, and decreasing if is greater than ^* _s,c, Moreover, ^* _s,c is increasing in the number of servers and, fors 1, ^* _i,c=1 p^* _s,c < 2.Research supported by Grant BD/645/90-RM from Junta Nacional de Investigação Científica e Tecnológica.On leave from: Departamento de Matemätica, Instituto Superior Técnico, Av. Rovisco Pais, 1096 Lisboa Codex, Portugal.     

Serre's Conjecture for Imaginary Quadratic Fields

We study an analog over an imaginary quadratic field K of Serre's conjecture for modular forms. Given a continuous irreducible representation :Gal(Q/K) GL₂(F_l) we ask if is modular. We give three examples of representations obtained by restriction of even representations of Gal(Q/Q). These representations appear to be modular when viewed as representations over K, as shown by the computer calculations described at the end of the paper.     

Smooth Measures and Regular Strongly Supermedian Kernels Generating Sub-Markovian Resolvents

In the context of a transient Borel right Markov process with a fixed excessive measure , we characterize the regular strongly supermedian kernels, producing smooth measures by the Revuz correspondence. In the case of the measures charging no -semipolar sets, this is the analytical counterpart of a probabilistic result of Revuz, Fukushima, and Getoor and Fitzsimmons, concerning the positive continuous additive functionals. We also consider the case of the measures charging no set that is both -polar and -negligible (U being the potential part of ), answering to a problem of Revuz.     

Approximation of the viability kernel

We study recursive inclusionsx ⁿ⁺¹ G(x ⁿ ). For instance, such systems appear for discrete finite-difference inclusionsx ⁿ⁺¹ G (x ⁿ) whereG :=1+F. The discrete viability kernel ofG , i.e., the largest discrete viability domain, can be an internal approximation of the viability kernel ofK underF. We study discrete and finite dynamical systems. In the Lipschitz case we get a generalization to differential inclusions of the Euler and Runge-Kutta methods. We prove first that the viability kernel ofK underF can be approached by a sequence of discrete viability kernels associated withx ⁿ⁺¹ (xⁿ) where (x) =x + F(x) + (ML/2) ². Secondly, we show that it can be approached by finite viability kernels associated withx _h ⁿ⁺¹ ( (x _h ⁿ⁺¹ ) +(h) X _h .     

Les classes de Chern des représentations galoisiennes complexes

Let F be a field, G _F its absolute Galois group, : G _FGL(C) a continuous complex representation of G _F and c _i() H_2i(F, Z) its Chern classes. We show, under a mild assumption on F. that c _i ()=0 for all i2. For general F, one has that 2c_i ()=0 for all i 2.

Cette dernière condition résulte en fait de la continuité de .     

The problem of calculation of change in density of rubber during temperature-frequency (time) reduction

A graphical method is proposed for determination of the coefficient of vertical displacement of curves of the frequency dependence of viscoelastic functions of polymers during temperature-frequency (time) reduction. With respect to temperature, the dependence of the quantity ₀/ (where ₀ and are the densities of the material at the reduction temperature T₀ and the initial temperature T, respectively) is allowed for by use of one constant, the volume coefficient of the rmal expansion .Scientific-Research Institute for the Plastics Industry, Moscow. Translated from Mekhanika Polimerov, No. 6, pp. 1136–1137, November–December, 1972.     

Some observations on the distribution of values of continued fractions

Summary There have been many studies of the values taken on by continued fractionsK(a _n /1) when its elements are all in a prescribed setE. The set of all values taken on is the limit regionV(E). It has been conjectured that the values inV(E), are taken on with varying probabilities even when the elementsa _n are uniformly distributed overE. In this article, we present the first concrete evidence that this is indeed so. We consider two types of element regions: (A)E is an interval on the real axis. Our best results are for intervals [–(1–), (1–)], 0 <1/2. (B)E is a disk in the complex plane defined byE={z:|z|(1–)}., 0<1/2.     

Relationship between the Hadamard Theorem on Three Disks and Certain Problems of Polynomial Approximation of Analytic Functions

By using the classical Hadamard theorem, we obtain an exact (in a certain sense) inequality for the best polynomial approximations of an analytic function f(z) from the Hardy space H _p, p 1, in disks of radii , ₁, and ₂, 0 < ₁ < < ₂ < 1.     

The Whitehead Group of the Novikov Ring

The Bass–Heller–Swan–Farrell–Hsiang–Siebenmann decomposition of the Whitehead group K ₁(A[z,z^-1]) of a twisted Laurent polynomial extension A[z,z^-1] of a ring A is generalized to a decomposition of the Whitehead group K ₁(A((z))) of a twisted Novikov ring of power series A((z))=A[[z]][z^-1]. The decomposition involves a summand W₁(A, ) which is an Abelian quotient of the multiplicative group W(A,) of Witt vectors 1+a₁z+a₂z²+ ··· A[[z]]. An example is constructed to show that in general the natural surjection W(A, )^ab W₁(A, ) is not an isomorphism.     

A Comparison of Two Spaces of Generalized White Noise Functionals

In the literature (see [5, 6, 8]) there are two families of spaces called Kondratiev spaces: (_c)^± and (S _c)^± for 0 1. We investigate the relation between the spaces and show that they are topologically isomorphic when (^d) L2 (^d) (^d) is the underlying Gel'fand triple for (_c)^±. In this case we also give the explicit relation between the S-transform and -transform on (_c)^-1 and (S _c)^-1, respectively.     

On recurrence

Summary LetT be a non-singular ergodic automorphism of a Lebesgue space (X,L,) and letf: X be a measurable function. We define the notion of recurrence of such a functionf and introduce the recurrence setR(f)={:f– is recurrent}. If , then R()={0}, but in general recurrence sets can be very complicated. We prove various conditions for a number to lie in R(f) and, more generally, forR(f) to be non-empty. The results in this paper have applications to the theory of random walks with stationary increments.